Question

the fifth term of a geometric sequence is 781.25. each term of the sequence is \\(\frac{1}{5}\\) of the value of the following term.

which recursive formula represents the situation?

\\(a_n = 5a_{n-1}; a_1 = 1.25\\)

\\(a_n = \frac{1}{5}a_{n-1}; a_1 = 488,281.25\\)

\\(a_n = 5a_{n-1}; a_1 = 488,281.25\\)

\\(a_n = \frac{1}{5}a_{n-1}; a_1 = 1.25\\)

Explanation:

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<reasoning>

Determine the common ratio \(r\)

\[

$$\begin{aligned} a_n &= \frac{1}{5} a_{n+1} \\ a_{n+1} &= 5 a_n \\ r &= 5 \\ a_n &= 5 a_{n-1} \end{aligned}$$

\]

Calculate the first term \(a_1\)

\[

$$\begin{aligned} a_5 &= a_1 \cdot r^4 \\ 781.25 &= a_1 \cdot 5^4 \\ 781.25 &= a_1 \cdot 625 \\ a_1 &= \frac{781.25}{625} = 1.25 \end{aligned}$$

\]

Match with the correct recursive formula

\[
a_n = 5 a_{n-1}; a_1 = 1.25
\]
</reasoning>

<answer>
<mcq-correct>(A) \(a_n = 5a_{n-1}; a_1 = 1.25\)</mcq-correct>
<mcq-option>(B) \(a_n = \frac{1}{5}a_{n-1}; a_1 = 488,281.25\)</mcq-option>
<mcq-option>(C) \(a_n = 5a_{n-1}; a_1 = 488,281.25\)</mcq-option>
<mcq-option>(D) \(a_n = \frac{1}{5}a_{n-1}; a_1 = 1.25\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Determine the common ratio \(r\)

\[

$$\begin{aligned} a_n &= \frac{1}{5} a_{n+1} \\ a_{n+1} &= 5 a_n \\ r &= 5 \\ a_n &= 5 a_{n-1} \end{aligned}$$

\]

Calculate the first term \(a_1\)

\[

$$\begin{aligned} a_5 &= a_1 \cdot r^4 \\ 781.25 &= a_1 \cdot 5^4 \\ 781.25 &= a_1 \cdot 625 \\ a_1 &= \frac{781.25}{625} = 1.25 \end{aligned}$$

\]

Match with the correct recursive formula

\[
a_n = 5 a_{n-1}; a_1 = 1.25
\]
</reasoning>

<answer>
<mcq-correct>(A) \(a_n = 5a_{n-1}; a_1 = 1.25\)</mcq-correct>
<mcq-option>(B) \(a_n = \frac{1}{5}a_{n-1}; a_1 = 488,281.25\)</mcq-option>
<mcq-option>(C) \(a_n = 5a_{n-1}; a_1 = 488,281.25\)</mcq-option>
<mcq-option>(D) \(a_n = \frac{1}{5}a_{n-1}; a_1 = 1.25\)</mcq-option>
</answer>

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